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Varieties of Distributed Knowledge

Rustam Galimullin, Louwe B. Kuijer

TL;DR

Varieties of Distributed Knowledge develops a uniform framework to classify distributed knowledge by 12 variant definitions, each determined by form of information, amount, order, and learning scope. It formalizes a general semantics $\models_\tau$ for each variant, links several variants to traditional intersection and full-communication notions, and analyzes which variants imply or are equivalent to one another. The work reveals that single-formula variants collapse to the same expressivity, while set-of-formulas variants can remain strictly stronger, yielding a nuanced map of distributed-knowledge expressivity across ordinal-length communications. This creates a comprehensive landscape for understanding how different information-sharing protocols affect what a group can collectively infer, with implications for the axiomatization and dynamic extensions of distributed knowledge in epistemic logic.

Abstract

Distributed knowledge is one of the better known group knowledge modalities. While its intuitive idea is relatively clear, there is ample room for interpretation of details. We investigate 12 definitions of distributed knowledge that differ from each other in the kinds of information sharing the agents can perform in order to achieve shared mutual knowledge of a proposition. We then show which kinds of distributed knowledge are equivalent, and which kinds imply each other, i.e., for any two variants $τ_1$ and $τ_2$ of distributed knowledge we show whether a proposition $φ$ being distributed knowledge under definition $τ_1$ implies that $φ$ is distributed knowledge under definition $τ_2$.

Varieties of Distributed Knowledge

TL;DR

Varieties of Distributed Knowledge develops a uniform framework to classify distributed knowledge by 12 variant definitions, each determined by form of information, amount, order, and learning scope. It formalizes a general semantics for each variant, links several variants to traditional intersection and full-communication notions, and analyzes which variants imply or are equivalent to one another. The work reveals that single-formula variants collapse to the same expressivity, while set-of-formulas variants can remain strictly stronger, yielding a nuanced map of distributed-knowledge expressivity across ordinal-length communications. This creates a comprehensive landscape for understanding how different information-sharing protocols affect what a group can collectively infer, with implications for the axiomatization and dynamic extensions of distributed knowledge in epistemic logic.

Abstract

Distributed knowledge is one of the better known group knowledge modalities. While its intuitive idea is relatively clear, there is ample room for interpretation of details. We investigate 12 definitions of distributed knowledge that differ from each other in the kinds of information sharing the agents can perform in order to achieve shared mutual knowledge of a proposition. We then show which kinds of distributed knowledge are equivalent, and which kinds imply each other, i.e., for any two variants and of distributed knowledge we show whether a proposition being distributed knowledge under definition implies that is distributed knowledge under definition .
Paper Structure (19 sections, 16 theorems, 14 equations, 1 figure)

This paper contains 19 sections, 16 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic Definitions
  3. Varieties of distributed knowledge
  4. Forms of information
  5. Amount of information
  6. Order and turn-taking
  7. Collective or individual success
  8. Equivalence among variants
  9. Semantics for distributed knowledge
  10. Non-linguistic sharing
  11. Simultaneous sharing of formulas
  12. Taking turns
  13. Connections to the traditional definitions
  14. Relative Strength
  15. Single formula
  16. ...and 4 more sections

Key Result

Theorem 1

Given $\mathcal{M},s$ and $\mathcal{N},t$, if $\mathcal{M},s \approx_Q \mathcal{N},t$, then for all $\varphi \in \mathcal{L}_0$ that include atoms only from $Q$, we have that $\mathcal{M},s \models \varphi$ if and only if $\mathcal{N},t \models \varphi$.

Figures (1)

  • Figure 1: The expressivity landscape of distributed knowledge, where 'NL' stands for 'non-linguistic sharing', 'Seq$\Omega$' denotes unlimited sequential sharing, 'Seq$\omega$' stands for sequential sharing limited to ordinal $\omega$, and 'Sim' denotes simultaneous sharing. Equivalent variations of distributed knowledge are enclosed in a box. Arrows point from stronger variants to weaker ones. Some arrows that follow from transitivity have been omitted for the sake of clarity.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Theorem 1
  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • ...and 23 more